Draft – July 17th, 2009 Math Grade 4
Important Note: The NC State Board of Education mandated that the adopted 2008 K-5 mathematics curriculum be revised using the same Essential Standards lens as A Framework For Change required of all standards. While this comes very quickly on the heels the recently adopted standards, this revision is important to ensure alignment and to build a sound, consistent mathematics program from Kindergarten through 12th grade. The big ideas in the 2008 standards are still present but may be reworded and, in some cases, moved to a different strand or grade level in these drafts. While writing these standards, the Revised Bloom's Taxonomy is being used to ensure uniformity and consistency in language and ensure the rigor of the standards.
Essential Standards • Math Grade 4
Number and Operations
4.N.1 Understand the value of whole numbers and decimal representations from 0.01 to 100,000.
4.N.2 Use a variety of strategies (including algorithms) to solve problems involving multi-digit addition, subtraction, multiplication and division of whole
numbers.
4.N.3 Understand the concept of equivalence with models as it applies to fractions, improper fractions, mixed numbers and decimals.
4.N.4 Use area and length models to represent addition and subtraction of fractions (with like denominators) and decimals.
Algebra
4.A.1 Use mathematical properties to examine numerical relationships and solve mathematical problems.
4.A.2 Use models to write multi-step equations and one-step inequalities with variables.
4.A.3 Analyze numeric and non-numeric patterns to identify rules that describe the pattern.
Geometry
4.G.1 Understand the concept of symmetry (line, rotational, both, neither) and its relationship to transformations (including reflections, translations and
rotations) as they apply to two-dimensional shapes.
Measurement
4.M.1 Use customary units to measure length, weight, capacity and temperature to solve problems.
4.M.2 Understand the relationship between area and perimeter of polygons.
Statistics and Probability
4.S.1 Interpret data from investigations involving one or two sets of data.
4.S.2 Predict the outcomes of simple probability experiments.
4th Grade Number and OperationMathematical language and symbols that students should use and understand at this grade level:
value, place value, represent, benchmark(s), expanded form, standard form, word form, picture form, sum, addend, difference, fraction, numerator, denominator,
mixed numbers, improper fractions, equivalent, halves, fourths, eighths, thirds, sixths, twelfths, fifths, tenths, hundredths, estimate, about, approximately, reasonable, justify, array, multiplication (×), factors, products, multiple, division (÷), quotients, remainders, dividend, divisor, less than (<), greater than (>), equal (=), not equal
( ≠), decimal, square numbers, prime, composite, ratio
Essential Standards
/Clarifying Objectives
/ Assessment Prototypes1
2
3
4
5
6
7
8
4.N.1 Understand the value of whole numbers and decimal representations from 0.01 to 100,000. /
4.N.1.1 Represent whole numbers and decimals using models, words and numbers (symbolic).
4.N.1.2 Compare sets of two to six numbers, arranging them from least to greatest or greatest to least.
4.N.1.3 Illustrate the place value structure of decimals and whole numbers when multiplying and dividing by 10. / 4.N.1 The sum of the digits of a three-digit mystery number is 14. The hundreds digit is twice the ones digit, and the tens digit is half of the ones digit. What is the mystery number? (constructed response) solution: 824
4.N.1 Use your calculator to count by 1,000 to one hundred thousand. If you count out loud, how many numbers will you say? Begin this way, “ten thousand, twenty thousand, thirty thousand…”(performance task)
4.N.1.1 Jim is counting the dimes in his bank to see how many dollars he has. He puts the dimes into piles of 10 since he knows that 10 dimes equals 1 dollar. Jim knows that 1 dime is one-tenth of a dollar. Write the value of a dime as a fraction and as a decimal. (constructed response) solution: fraction: and decimal fraction 0.01
1. What does the denominator mean in the fraction?
2. What does the numerator mean in the fraction?
3. In the decimal fraction 0.1 what does the zero mean?
4. If he had 235 dimes, write the value of this amount using decimal form.
4.N.1.1 Using the metric model below, place your pencil point on 3.5 cm, 2.8cm, 0.7 cm, etc… Does your partner agree with you? Explain how this model is different and similar to a number line? (performance task)
4.N.1.2 Using 5 number cards form two different 5-digit numbers. Use the numbers to generate a comparative statement to share with your partner, such as 32,786 32,687. (performance task)
4.N.1.2 Given distances of cities in the U.S. from Raleigh, put them in order of closest to farthest from Raleigh. (constructed response)
4.N.1.3 Find the following products. Look for a pattern. (constructed response)
3 x 8 =
30 x 8 =
3 x 80 =
30 x 80 =
300 x 80 =
30 x 800 =
300 x 800 =
1. What pattern did you see in the equations above?
2. Kirk said he could multiply 30 x 30 in his head easily. What method do you think Kirk is using? What is 30 x 30?
3. Kirk saw that for every zero in the factor, there is a zero in the product. Do you agree? Explain.
4. Kirk says multiplying 60 x 5000 is tricky. What is 60 x 5000? Why would Kirk say it is tricky?
5. N x 40 = 200
6. N x 10 = 2000
7. N x 50 = 2000
8. N x 80 = 4000
4.N.2 Use a variety of strategies (including algorithms) to solve problems involving multi-digit addition, subtraction, multiplication and division of whole numbers. / 4.N.2.1 Use strategies to develop fluency for multiplication with two-digit by one-digit and two-digit by two-digit numbers.
4.N.2.2 Use strategies with three-digit by one-digit division with and without remainders to develop fluency.
4.N.2.3 Use estimation of whole number operations in meaningful contexts to justify the reasonableness of the solution.
4.N.2.4 Use area models to understand multiplication and related concepts such as factors, multiples (including squares), primes and composite numbers. / 4.N.2.1 What number is missing in the computation below? Be ready to explain your solution. (constructed response) Students might complete the computation algorithmically, or they might draw a picture to illustrate the computation. solution:410
41
x 13
123
???
533
4.N.2.1 Sarah was asked to multiply 36 x 24. Sarah did the following.
6 x 4 = 24,
30 x 4 = 120
6 x 20 = 120
30 x 20 = 600
600 + 120 + 120 +24 = 864.
Is Sarah’s method correct? Why or why not? (constructed response)
4.N.2.2 There are 99 pizzas for the banquet. If the pizzas are shared evenly among 8 tables, how much pizza is put at each table? (constructed response)
4.N.2.2 The store packages chicken nuggets in different ways. Mrs. Rutherford buys 21 packs of 12 nuggets. If the same number of chicken nuggets is packed in bags of 9 nuggets how many bags would be needed? (constructed response) solution:21 packs times 12 equals 252. 252 divided by 9 equals 28 bags. Therefore 28 bags of chicken nuggets would be needed.
4.N.2.3 John went to the sports equipment store and priced the cost of Louisville Slugger baseball bats. He wrote down the cost, but then accidentally spilled milk on the paper on which it was written. Some of the digits were smeared: $3¨¨.75. If he wants to buy 5 bats, which amount of money would be closest to the actual cost? Be ready to explain your choice with your class. (multiple choice) This gets at the notion that estimation is different from “computing and then rounding”. solution: c
a. $200,000
b. $20,000
c. $2,000
d. $200
4.N.2.3 I’m thinking of a number. When I multiply that number by 40 the product is a little more than 2,000. Estimate the number I’m thinking of. (constructed response) solution: about 50
4.N.2.3 I’m thinking of a number. When I divide that number by 25 the quotient is very close to 30. Estimate the number I’m thinking of. (constructed response) solution: about 750
4.N.2.4 If an array model of 74 x 3 looks like:
70 4
210 / 12
What would an array model of 74 x 38 look like? (constructed response)
possible solution:
70 4
2100 /
120
560 / 32
4.N.2.4 If the sides of the rectangle above are each whole numbers, and if the shorter side is longer than 10 inches, what might the dimensions of the rectangle be? Be ready to explain how you got your answer. (constructed response) solution: 12 x 50, 15 x 40, 20 x 30, 24 x 25; 10 x 60 is not an answer, since the shorter side must be longer than 10 inches.
4.N.3 Understand the concept of equivalence with models as it applies to fractions, improper fractions, mixed numbers and decimals. / 4.N.3.1 Identify equivalent fractions (using halves, fourths, eights; thirds, sixths, twelfths and fifths, tenths, hundredths)
4.N.3.2 Compare fractions, decimals and mixed numbers using models, benchmarks (e.g. 0, 1/2, 1, 1.5, 2) and reasoning. / 4.N.3.1 A candy bar has 12 sections. John has of a candy bar. Sara said he had of a candy bar. Is she correct? Explain how you know. What are some other ways John could identify his portion of the candy bar? (constructed response)
4.N.3.1 Kyle’s mom brought a pie for the class picnic. It was cut into 6 pieces. Gracie’s dad also brought a pie, but it was cut into 12 pieces. At the end of the picnic of Kyle’s pie was left and of Gracie’s pie was left. If the pies were the same size, who had more pie left over. Kyle’s mom or Gracie’s dad? Explain your reasoning. (constructed response)
4.N.3.1 Amy wants to purchase yard of ribbon. How many inches of ribbon must she cut? (constructed response) a ribbon and yard stick are continuous models, solution: = so = so She must cut 12 inches of ribbon.
4.N.3.1 Terry is helping his father pack a box of pencils for a fundraiser. The box holds pound of pencils. Each pencil weighs 1/16 of a pound. How many pencils can fit in the box? (constructed response) pencils and ounces are discrete models, solution: =so = so 8 pencisl can fit in a box.
4.N.3.1 Students make fraction strips by folding strips of paper, first fold in , then in half again to make etc…. (performance task)
1. Use your fraction strip to find a fraction that is equivalent to.
2. Write a number sentence to record the equivalent fractions.
3. Write three other fractions that are equivalent to
4. Explain the strategy you used to find the equivalent fractions.
4.N.3.2 Four swimmers race in the 100 meter freestyle Olympics. Peter swims the distance in 27.97 seconds. Bert swims the distance in 27.9 seconds. Mark swims the distance in 27.92 seconds. Hank swims in 27.91 seconds. Put them in order from first to last place. (constructed response)
possible model representation 27.9 - 1st place
27.91 - 2nd place
27.92 - 3rd place
27.97 - 4th place
27.97 27.9
27.92 27.91
8.3.2
8.3.3 Use a number line that has fractions 0, ½ and 1 marked and identify where these decimals would appear on the number line: .60, .25, .5. (constructed response)
8.3.4 Jodi has a stick that is ½ a foot long. In order to measure the height of her desk she realizes that the desk is 9 sticks high. How high is the desk in terms of feet? Explain your work. (constructed response)
8.3.5 The fourth grade class was making lemonade for the class picnic. The recipe they have says that 6 lemons and 9 cups of water makes great lemonade, but that won’t make enough for the picnic. Fill in the following table, so the class can make different amounts of lemonade that will taste the same. (constructed response)
no. of lemons / no. of cups of water
6 / 9
What is the smallest ratio of lemons to water that will still use whole lemons? (constructed response)
4.N.3.3 Represent mixed numbers as improper fractions and improper fractions as mixed numbers.
4.N.3.4 Understand the equivalence of fractions as related to the size of the whole.
4.N.3.5 Represent pairs of equivalent ratios by composing and decomposing to include the smallest equivalent whole number ratio (ratio unit). / 4.N.3.3 Share with your classmates a strategy for rewriting 5 as (performance indicator)
4. N.3.3 Explain why the numbers in each pair are or are not equivalent: (performance indicator)
4 and 3 solution: yes, but they are not in simplest form
and 1 solution: yes
4.N.3.4 Mary used a 12 x 12 grid to represent 1 and Jane used a 10 x 10 grid to represent 1. Each girl shaded little squares to show . How many little squares did Mary shade? How many little squares did Jane shade? Why did they need to shade different numbers of little squares? (constructed response) solution: Mary shaded 36 little squares; Jane shaded 25 little squares. The total number of little squares is different in the two grids, so of each total number is different.
Mary’s grid Janet’s grid
4.N.3.4 Use patterns blocks. (performance task)
1. If a red trapezoid is one whole, which block shows ?
2. If the blue rhombus is , which block shows one whole?
3. If the red trapezoid is one whole, which block shows ?